Multivariable calculus - Limits

Mar 2019
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Show that the function f(x,y)=y/(x-y) for x→0, y→0, can take any limit.
Construct the sequences { f(xn, yn } with (xn,yn)→(0, 0) in such way that the lim n→∞ f(xn,yn) is 3,2,1,0,−2.
Hint: yn=kxn.

I am not sure whether I am right, but I did the following:

f(x,y) = kxn/(xn−kxn) = kxn/(xn(1−k)) = k/(1−k)

k/(1−k) = A

After a few steps I have obtained:

k = A/(1+A), A ≠ −1

So, the function can take any limit except -1.

Now we have: A {3,2,1,0,−2}

If A=3:

k=3/4, yn=3/4xn

So lim n→∞ (3/4xn)/(1/4xn)=3.

For A = 2, 1, 0 and −2, I did the same.

I am not sure whether this is the end of the exercise. I don't understand what exactly in this context is meant by the sequence.

I am also not sure whether I should write lim (xn, yn)→(0,0) (3/4xn)/(1/4xn)=3 instead of lim n→∞ (3/4xn)/(1/4xn)=3 or just xn→0, because I have just xn.

Any help is appreciated.

Thanks in advance.
 
Last edited by a moderator:
Dec 2015
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If you change the order of limits you get \(\displaystyle -1=\lim_{x\rightarrow 0} \frac{0}{-x}\) which is not true .
 
Last edited:

v8archie

Math Team
Dec 2013
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So, the function can take any limit except -1.
You were asked to show that it can take any limit. How are you going to get -1? Hint: approach along the $y$ axis.
I am also not sure if I should write lim (xn, yn)→(0,0) (3/4xn)/(1/4xn)=3 instead of lim n→∞ (3/4xn)/(1/4xn)=3 or just xn→0, because I have just xn.
Having shown, for example, that with $y_n=\frac34 x_n$ you get $\displaystyle \lim_{n \to \infty} f(x_n,y_n) = 3$, you should interpret what $y_n=\frac34x_n$ means in terms of the approach to $(0,0)$. In this case you are approaching the origin along the straight line $y = \frac34x$. This is the answer that I consider most appropriate, although the talk of sequences may mean that your teacher has something else in mind.
 
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v8archie

Math Team
Dec 2013
7,712
2,682
Colombia
If you change the order of limits you get \(\displaystyle -1=\lim_{x\rightarrow 0} \frac{0}{-x}\) which is not true .
There is no "order of limits" here. The idea is simply to determine a path of approach to $(0,0)$ that delivers the appropriate limit.
 
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